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Unformatted text preview: Complex Variables In the calculus of functions of a complex variable there are three fundamental tools, the same funda- mental tools as for real variables. Differentiation, Integration, and Power Series. I’ll first introduce all three in the context of complex variables, then show the relations between them. The applications of the subject will form the major part of the chapter. 14.1 Differentiation When you try to differentiate a continuous function is it always differentiable? If it’s differentiable once is it differentiable again? The answer to both is no. Take the simple absolute value function of the real variable x . f ( x ) = | x | = x ( x ≥ )- x ( x < ) This has a derivative for all x except zero. The limit f ( x + Δ x )- f ( x ) Δ x-→ 1 ( x > )- 1 ( x < ) ? ( x = 0 ) (14 . 1) works for both x > and x < . If x = 0 however, you get a different result depending on whether Δ x → through positive or through negative values. If you integrate this function, Z x | x | dx = x 2 / 2 ( x ≥ )- x 2 / 2 ( x < ) the result has a derivative everywhere, including the origin, but you can’t differentiate it twice. A few more integrations and you can produce a function that you can differentiate 42 times but not 43. There are functions that are continuous but with no derivative anywhere. They’re harder* to construct, but if you grant their existence then you can repeat the preceding manipulation and create a function with any number of derivatives everywhere, but no more than that number anywhere. For a derivative to exist at a point, the limit Eq. ( 14.1 ) must have the same value whether you take the limit from the right or from the left. Extend the idea of differentiation to complex-valued functions of complex variables. Just change the letter x to the letter z = x + iy . Examine a function such as f ( z ) = z 2 = x 2- y 2 + 2 ixy or cos z = cos x cosh y + i sin x sinh y . Can you differentiate these (yes) and what does that mean? f ( z ) = lim Δ z → f ( z + Δ z )- f ( z ) Δ z = df dz (14 . 2) is the appropriate definition, but for it to exist there are even more restrictions than in the real case. For real functions you have to get the same limit as Δ x → whether you take the limit from the right * Weierstrass surprised the world of mathematics with ∑ ∞ a k cos( b k x ) . If a < 1 while ab > 1 this is continuous but has no derivative anywhere. This statement is much more difficult to prove than it looks. James Nearing, University of Miami 1 14—Complex Variables 2 or from the left. In the complex case there are an infinite number of directions through which Δ z can approach zero and you must get the same answer from all directions. This is such a strong restriction that it isn’t obvious that any function has a derivative. To reassure you that I’m not talking about an empty set, differentiate z 2 ....
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This note was uploaded on 01/30/2012 for the course PHYS 315 taught by Professor Nearing during the Fall '08 term at University of Miami.
- Fall '08